The comments to Arnold’s problem 1989–11 [see [Arnold2004]] contain
the following conjecture:

Let us have an elliptic curve with a zero self-intersection
index in a complex surface. Then, in a neighborhood of this curve,
there exists a foliation tangent to the curve.

Problem 1989–11 itself concerns holomorphic classification of
neighborhoods of compact complex curves in complex surfaces. Its
prehistory in the theory of dynamical systems is as follows.

Consider a saddle of a germ of an automorphism of $\mathbb{C}^{2}$. It
has two separatrices. On any one of them, the automorphism can be
reduced to a linear normal form. In the linearizing coordinate, it
maps a small zero-centered circle to a circle of some other radius.
These two circles bound an annulus. If we identify the circles using
our automorphism, the annulus becomes an elliptic curve. If we make
the same identification in a neighborhood of the annulus, this
neighborhood becomes a neighborhood of this elliptic curve in some
surface.

[Arnold1976] suggested a construction to relate
properties of the dynamical system to the properties of the complex
structure of the neighborhood11Instead of separatrices,
Arnold used invariant submanifolds, the existence of which had been
proved by [Pyartli1973] and [Bryuno1975]..
For example, holomorphic normal forms of such neighborhoods would
imply holomorphic normal forms of such automorphisms, compact
submanifolds of the neighborhood will become invariant submainfolds
of the automorphisms, etc.

Following this idea, [Arnold1997] ([Arnold1997], §27), studied
two-dimensional neighborhoods of elliptic curves, having a zero
self-intersection in these neighborhoods. Studying such
neighborhoods was continued by [Ilyashenko and Pyartli1979], [Ilyashenko1982] and by the author of this note in
[Mishustin1993]; [Mishustin1995].

Neighborhoods, studied by Arnold, appeared to be objects with
“small denominators” or having “dense resonances” or belonging to
“the Siegel case”. This means that, for a dense subset of linear
terms (the set of “resonant points”), geometry of objects radically
changes, in particular, the dimensions of spaces of normal forms of
finite jets of objects change. For such objects, the formal
normalizing transformations may diverge. For non-resonant linear
terms, smaller and smaller denominators appear in formal series
which may cause divergence.

Small denominators appear in many mathematical models, starting with
Poincare’s normal forms of differential equations. A typical result
for such models provides normal forms for “normally
incommensurable” objects, whose linear terms are “not too well
approximated by resonant ones” and delivers examples of divergence
of formal normalizing transformations for other cases [see, for
example, [Bryuno1971], Theorems II and III]. The resonant and
abnormally commensurable cases are much harder to describe, despite
lots of efforts. An exhaustive study of resonant germs of
authomorphisms of $\mathbb{C}$ was performed by [Voronin1981]. [Bryuno1971] constructed some partial
normal forms for differential equations of this kind.

The linear term of the neighborhood is defined by the normal bundle
of the curve in the surface. The normal bundle is resonant when some
its symmetric power is holomorphically trivial. The normal bundle is
normally incommensurable, when its $k$th symmetric power is no
closer to the trivial bundle than $C/k^{m}$ for suitable
$C\in\mathbb{R}^{+},m\in\mathbb{N}$ (not depending on $k$). For this
case, [Ilyashenko and Pyartli1979] proved that a
neighborhood of the curve in a surface is biholomorphically
equivalent to a neighborhood of the zero section in the space of the
normal bundle.

The conjecture in question attempted to go a little further and to
“decompose” a dynamic chaos to the “laminar” and “chaotic”
components for any linear term. The hypothesis was based on the
following two observations:

(a)

If the normal bundle is normally incommensurable, then the
foliation can be easily constructed in the space of the normal
bundle. Let the normal bundle be abnormally commensurable. Include
the neighborhood into a family of neighborhoods that deforms the
normal bundle. In dense resonant members of the family, a covering
curve will detach from the original curve. The detaching curve will
also be elliptic, with a zero self-intersection and a zero
intersection with the original curve. Its normal bundle will also be
deformed by the family; therefore, in dense subset of the family,
the covering of the detached curve will also detach, and so on. So,
we can expect rather a dense set of non-intersecting curves in a
neighborhood of the original curve. We can suppose that they are
compact leaves of some holomorphic foliation. The differentials of
recurrence maps of this foliation along cycles on the original curve
should be unitary, and this condition on foliation is necessary for
non-linearizable neighborhoods.

(b)

If we construct formal series of a foliation with unitary
differentials of recurrence maps, it turns out that, for every small
denominator, there will appear a small numerator conjugate to it.
Thus, all finite jets of the foliation do not tend to infinity in
resonant points, although they are not unique in resonances.

Arnold himself, though admitted this conjecture in the comment,
treated it rather skeptically, because “it seemed unlikely that the
neighborhoods could be constructed so easily”. He was right.

Consider the trivial bundle over an elliptic curve supplied with
coordinates $(r,\varphi)$ on fibers and the base respectively,
$\varphi\sim\varphi+2\pi\sim\varphi+\omega$. For the sake of
definiteness, let $\omega=2\pi i$.

Let $a(r,\varphi)$ be a function on a neighborhood of the zero
section, smooth with respect to $\varphi$, holomorphic with respect
to $r$, and vanishing for $r=0$. Let us define a nonstandard almost
complex structure $J$ in this neighborhood given in the basis $(\partial/\partial r,\partial/\partial\varphi,\partial/\partial\overline{r},%
\partial/\partial\overline{\varphi})$ by
the matrix

the operator $\overline{\partial}_{J}=(d+JdJ)/2$ has a
form of
$\overline{\partial}_{J}=\overline{\partial}+d\overline{\varphi}\cdot a(r,%
\varphi)\cdot\partial/\partial r$;

(c)

$\overline{\partial}_{J}^{2}=0$, because $a(r,\varphi)$ is
holomorphic with respect to $r$.

The latter means that the structure $J$ is integrable, and the
domain of $a(r,\varphi)$, equipped with the structure $J$, is a
complex surface. Since $a(0,\varphi)=0$, the structure $J$ coincides
with the standard structure on the zero section, hence the zero
section is the same elliptic curve, embedded in this surface.

Suppose that, in a neighborhood of the curve, there exists a
foliation, holomorphic with respect to the structure $J$ and tangent
to the curve. In a smaller neighborhood of the curve, this foliation
can be defined by the equation $dr=u(r,\varphi)d\varphi+v(r,\varphi)d\overline{\varphi}$ for suitable smooth $u(r,\varphi)$
and $v(r,\varphi)$.

The $J$-invariance of the foliation implies that
$v(r,\varphi)=a(r,\varphi)$.

Since the foliation is holomorphic on a two-dimensional complex
manifold, it is integrable. The Frobenius condition implies:

Equation (1) shows that $u(r,\varphi)$ is holomorphic with
respect to $r$. Since the foliation is tangent to the curve,
$u(0,\varphi)=0$. So we can expand $u$ and $v$ into series with
respect to $r$: $u(r,\varphi)=\sum_{k\geq 0}u_{k}(\varphi)r^{k+1}$ and
$v(r,\varphi)=\sum_{k\geq 0}v_{k}(\varphi)r^{k+1}$. Then (1)
implies, for every $k\geq 0$,

Let $a$ be of the following form:
$a(r,\varphi)=r^{2}+r^{3}(e^{\varphi-\overline{\varphi}}+e^{i\varphi+i\overline%
{\varphi}})$.
Then $v_{1}(\varphi)=1,v_{2}(\varphi)=e^{\varphi-\overline{\varphi}}+e^{i\varphi+i%
\overline{\varphi}}$, and $v_{k}=0$ for other $k$.

For $k=0$, we have $\partial u_{0}/\partial\overline{\varphi}=0$,
so $u_{0}(\varphi)=u_{000}$.

For $k=1$, we have $\partial u_{1}/\partial\overline{\varphi}=u_{000}$. This equation can be solved only if $u_{000}=0$, and then
$u_{1}(\varphi)=u_{100}$ and $u_{0}(\varphi)=0$.

For $k=2$, we have $\partial u_{2}/\partial\overline{\varphi}=\partial v_{2}/\partial\varphi=e^{%
\varphi-\overline{\varphi}}+ie^{i\varphi+i\overline{\varphi}}$, so
$u_{2}(\varphi)=-e^{\varphi-\overline{\varphi}}+e^{i\varphi+i\overline{\varphi}%
}+u_{200}$.

For $k=3$, we have $\partial u_{3}/\partial\overline{\varphi}=u_{1}v_{2}-u_{2}v_{1}=(u_{100}+1)e^{%
\varphi-\overline{\varphi}}+(u_{100}-1)e^{i\varphi+i\overline{\varphi}}-u_{200}$. Solvability of this
equation requires $u_{200}=0$, and then we have

Let us continue this reasoning by induction for $k>3$, neglecting
the terms of $e^{\varphi-\overline{\varphi}},e^{i\varphi+i\overline{\varphi}}$ of degree $\geq 2$. We obtain the
following induction step:

Now suppose that the series $\sum_{k\geq 0}u_{k}(\varphi)r^{k+1}$
converges in the closure of some domain $U_{\rho}:\{|r|<\rho\}$ for
small enough $\rho>0$. In the space $L^{2}(U_{\rho})$, corresponding
to the standard metric $|dr|^{2}+|d\varphi|^{2}$, homogenous terms
$u_{k}(\varphi)r^{k+1}$ are orthogonal to each other. Besides, in
(3), the terms for $e^{\varphi-\overline{\varphi}}$, the
terms for $e^{i\varphi+i\overline{\varphi}}$, and the sum of terms
of degree $\geq 2$ of $e^{\varphi-\overline{\varphi}}$ and $e^{i\varphi+i\overline{\varphi}}$ are all orthogonal to each other.
Thus,

Coefficients $(u_{100}+1)$ and $(u_{100}-1)$ cannot both be zero;
so, in view of the factor $(k-2)!$, the latter series diverges for
any $\rho>0$.

Thus, for the trivial normal bundle, the counterexample has been
constructed. It can be easily modified for any resonant bundle. The
author believes that it is possible to adapt it to abnormally
commensurable normal bundles.